In these lessons, we will learn

- how to graph of cubic functions by plotting points.
- how to graph cubic functions of the form y = a(x − h)
^{3}+ k.

**Related Pages**

Factor Theorem

Solving Quadratic Equations

Graphs Of Quadratic Functions

More Algebra Lessons

The general form of a **cubic function** is
y = ax^{3} + bx + cx + d where a , b, c and d are real numbers and a is not zero.

We can graph cubic functions by plotting points.

**Example:**

Draw the graph of y = x^{3} + 3 for –3 ≤ x ≤ 3. Use your graph to find

a) the value of y when x = 2.5

b) the value of x when y = –15

**Solution:**

x | –3 | –2 | –1 | 0 | 1 | 2 | 3 |

y | –24 | –5 | 2 | 3 | 4 | 11 | 30 |

a) When x = 2.5, y ≈ 18.6

b) When y = –15, x ≈–2.6

**Example:**

Plot the graph of y = x^{3} – 9x + 5 for –4 ≤ x ≤ 4 and use your
graph to find:

a) the value of y when x = 1.6

b) the value of x when y = 12

**Solution:**

x | –4 | –3 | –2 | –1 | 0 | 1 | 2 | 3 | 4 |

y | –23 | 5 | 15 | 13 | 5 | –3 | –5 | 5 | 33 |

a) When x = 1.6, y ≈ –5.3

b) When y = 12, x ≈ –0.8, or x ≈ –2.5

**How to graph cubic functions using a calculator or technology?**

**How to graph a cubic or degree 3 polynomial function by completing a table of values?**

Complete the table using the function rule

f(x) = x^{3} - 4x and graph the function.

We can graph cubic functions by transforming the basic cubic graph. The basic cubic graph
is y = x^{3}.

For the function of the form y = a(x − h)^{3} + k.

If k > 0, the graph shifts k units up; if k < 0, the graph shifts k units down.

If h > 0, the graph shifts h units to the right; if h < 0, the graph shifts h units left.

If a < 0, the graph is flipped.

**How to graph cubic functions by writing the function in the form y = a(x − h) ^{3} + k?**

**Example:**

Graph \(y = - \frac{1}{2}{\left( {x + 4} \right)^3} + 5\)

**How to graph a Transformation of a Cubic Function?**

**Example:**

Sketch the graph of \(f(x) = - \frac{3}{2}{\left( {x + 2} \right)^3} - 3\)

**Graphing cubics using end behavior, inverted cubic, vertical shift, horizontal shift, and combined shifts**

**Graphing cubics using combined shifts, vertical stretch**

**Cubic Equation Solver**

This Cubic Equation calculator will solve the given cubic equation. Use it to check your answers.

Try the free Mathway calculator and
problem solver below to practice various math topics. Try the given examples, or type in your own
problem and check your answer with the step-by-step explanations.

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